Method and apparatus utilizing nuclear magnetic resonance measurements for estimating residual carbon dioxide saturation in aquifers

ABSTRACT

Percolation theory is applied to establish a connection between magnetization decay of nuclear magnetic resonance (NMR) measurements and residual carbon dioxide saturation (S cr ). As a result, estimations of S cr  are obtained through use of an NMR tool in a formation and appropriate processing. Data may be displayed as a log.

CROSS-REFERENCE TO RELATED APPLICATION

This application claims priority as a continuation application of U.S. patent application Ser. No. 12/909,116, filed on Oct. 21, 2010 with the same title that will issue as U.S. Pat. No. 8,452,539. The U.S. patent application Ser. No. 12/909,116 is incorporated by reference herein.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates broadly to estimating residual carbon dioxide (CO₂) saturation in aquifers. More particularly, this invention relates to estimating residual CO₂ saturation (S_(cr)) from nuclear magnetic resonance (NMR) measurements obtained by a logging tool.

2. State of the Art

Elevated carbon dioxide concentration in the atmosphere is widely accepted as a contributor to global climate change. Carbon capture and sequestration (CCS) is one of the pursued technologies to reduce atmospheric accumulation of CO₂.

Suitability of a carbon dioxide geological storage site is commonly characterized by three metrics: capacity, infectivity and containment. Evaluation of these three performance measures at early stages of a CO₂ storage project relies largely upon seismic and well characterization, reservoir modeling and simulation. Petrophysical properties such as porosity, permeability and residual saturation of aqueous and CO₂-rich phases describe the target formation zone, and serve as inputs for simulation models. The estimates for these properties are usually inferred from wireline measurements. Porosity is usually estimated based on neutron scattering and density measurements, while permeability is commonly inferred from NMR measurements. See, Timur, A. “Pulsed Nuclear Magnetic Resonance Studies of Porosity, Moveable Fluid, and Permeability of Sandstones,” Journal of Petroleum Technology, 21:775-786 (1969); Kenyon, W. E., et al., “A Three-part Study of NMR Longitudinal Relaxation Properties of Water-Saturated Sandstones,” SPE Formation Evaluation, 3:622-636 (1988); Kenyon, W. E., et al., “Erratum,” SPE Formation Evaluation, 4:8 (1989). In practice, NMR based inference is not absolute and requires zonal calibration. Methods for evaluating surface relaxivity from stationary formation tests to calibrate NMR logs are proposed in U.S. Pat. No. 7,221,158 to Ramakrishnan which is hereby incorporated by reference herein in its entirety.

In geological storage, brine displaced by CO₂ counter-imbibes to form trapped or residual CO₂. This refers to the part of the CO₂-rich phase disconnected from the rest of the phase exhibiting pressure continuity. Unlike oil wells, in geological storage sites, CO₂ is not present during drilling. Therefore drilling fluid filtrate invasion consists of a single phase displacement, and an estimate of S_(cr) cannot be obtained. While estimations of residual saturations can be obtained as part of an advanced core analysis which is conducted in the lab through displacement experiments, laboratory methods are laborious and are available only at formation locations and depths which have been subjected to coring. As will be appreciated by those skilled in the art, formation coring is slow and expensive, and provides information for only the specific coring locations. Given the desire to rapidly develop geological CO₂ storage worldwide, reliance on coring is not a suitable option.

SUMMARY OF THE INVENTION

In accord with the present invention, residual carbon dioxide saturation is estimated from NMR measurements obtained by a logging tool.

According to one aspect of the invention, utilizing NMR measurements, residual carbon dioxide saturations S_(cr) are estimated as a function of depth in a borehole, and a continuous log may be presented.

According to one embodiment, the porous medium of the formation is approximated by a parameterized pore-level percolation model. The values of the model parameters are inferred from petrophysical measurements that also relate to the same parameters, or are estimated for known types of formations. The remaining petrophysical properties of the modeled medium, e.g., residual CO₂ saturation, are then calculated from the pore level model. Thus, it is expected that the inferences and the measurements of reservoir properties will be self-consistent within the percolation framework. More particularly, percolation theory is applied to establish a connection between magnetization decay of NMR measurements and residual CO₂ saturation, and algorithms are provided for calculating S_(cr) based on available petrophysical data.

In one embodiment, S_(cr) is estimated from NMR measurements in the absence of other data. In another embodiment S_(cr) is estimated from NMR measurements in view of independent residual water saturation (S_(wr)) determinations. In yet another embodiment, S_(cr) is estimated from NMR measurements in view of independent permeability measurements. In all three embodiments, continuous logs of S_(cr) may be generated.

Additional objects and advantages of the invention will become apparent to those skilled in the art upon reference to the detailed description taken in conjunction with the provided figures.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a flow chart of a first embodiment of a method for generating residual carbon dioxide saturation estimations as a function of depth.

FIG. 2 is a flow chart of a second embodiment of a method for generating residual carbon dioxide saturation estimations as a function of depth.

FIG. 3 is a flow chart of a third embodiment of a method for generating residual carbon dioxide saturation estimations as a function of depth.

FIG. 4 is a schematic diagram of a borehole tool in a formation and a system for generating logs of residual carbon dioxide saturation in the formation utilizing any of the methods of FIGS. 1 through 3.

FIG. 5 a is a log showing residual carbon dioxide saturation as a function of formation depth.

FIG. 5 b is a log showing a representation of a probability density function of residual carbon dioxide saturation as a function of formation depth.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

Prior to turning to the drawings, an understanding of the theoretical underpinnings of the methods of the invention is desirable.

The theoretical basis for the methodology of the invention is a percolation network model. Based on assumptions (e.g., a water-wet medium, a one-to-one size correspondence between pore throats and pore body), the pore size NMR relaxation distribution may be related to permeability and residual saturations of both the wetting and non-wetting phases using percolation theory. The first step to carry this out is to relate the observed magnetization relaxation to a transverse relaxation time distribution.

As in previously incorporated U.S. Pat. No. 7,221,158, the transverse magnetization decay from NMR measurements is denoted as M(t), and it is assumed that the formation pore space can be represented by a probability density function of pore sizes, with relaxation in each pore being surface relaxivity controlled. Assuming a sufficiently short echo spacing in the NMR measurements (e.g., CPMG echo spacing used in current NMR tools such as the CMR tool of Schlumberger is 0.2 ms), the diffusion in the observed spin-spin relaxation may be ignored. Thus, the observed relaxation time (T_(2o)) associated with each pore is defined according to

$\begin{matrix} {{\frac{1}{T_{{2\; o}\;}} = {\frac{1}{T_{2b}} + \frac{\rho}{l}}},} & (1) \end{matrix}$

where T_(2b) is the bulk transverse relaxation time of the fluid (for water, T_(2b) is in the range of 2.5-3 s), l is the volume to surface area ratio of the pore, and ρ is surface relaxivity. If g_(l)(l) is the volume probability density function of l, then, by definition,

∫₀ ^(∞) g _(l)(l)dl=1  (2)

Combining Eq. 1 and Eq. 2, the spin-spin relaxation probability density function with respect to T_(2o) is defined by

$\begin{matrix} {{\left. {\int_{0}^{T_{2b}}{{g_{o}\left( T_{2o} \right)}{T_{2o}}}} \right) = 1},{where}} & (3) \\ {{T_{2o} = {\frac{l}{l + {\rho \; T_{2b}}}T_{2b}}};{{g_{o}\left( T_{2o} \right)} = {\frac{1}{\rho}\left( \frac{l}{T_{2o}} \right)^{2}{g_{l}(l)}}}} & (4) \end{matrix}$

Since the transverse magnetization relaxation in each pore decays as exp(−t/T_(2o)), the total observed magnetization measured by an NMR tool represents volumetric superposition of responses from all pores according to:

M(t)=φ∫₀ ^(T) ^(2b) g _(o)(T _(2o))exp(−t/T _(2o))dT _(2o)  (5)

U.S. Pat. No. 5,363,041 to Sezginer which is hereby incorporated by reference herein in its entirety discloses a computational procedure to determine g_(o)(T_(2o)) from M(t) using singular value decomposition.

Among the properties governing transport equations in porous media, residual saturations of the wetting and non-wetting phases are of particular interest. According to Ramakrishnan, T. S, and Wasan, D. T., “Two-phase Distribution in Porous Media: An Application of Percolation Theory,” Int. J. Multiphase Flow, 12(3):357-388 (1986), the residual wetting phase is trapped in small pores that are not accessed from the boundary by a connected path of pore filled wetting phase. The residual wetting phase may however be connected to the boundary through thin films or through roughness filling fluid. The wetting phase trapping may be characterized by the cumulative relaxation response from the portion of pores up to certain critical characteristic pore-size l_(wc). In terms of spin-spin relaxation time, the residual wetting phase saturation S_(wr) can be determined from

S _(wr)∫₀ ^(T) ^(2wc) g _(o)(T _(2o))dT _(2o),  (6)

where T_(2wc) is the critical spin-spin relaxation time related to the pore-size characterized by l_(wc) via Eq. 1. The critical l_(wc) is quantified below.

Unlike the wetting phase, the non-wetting phase is trapped when disconnected from the rest of the flowing phase and has no thin film interconnections. The isolated blobs are separated or surrounded by the wetting phase. As suggested by Mohanty, K. K., “Fluids in Porous Media: Two-Phase Distribution and Flow,” Ph.D. Thesis, University of Minnesota (1981) and Ramakrishnan, T. S, and Wasan, D. T., “Two-phase Distribution in Porous Media An Application of Percolation Theory,” Int. J. Multiphase Flow, 12(3):357-388 (1986), two different entrapment mechanisms exist. The first one is a snap-off event and is caused by the instability of the wetting phase thin film when the pore body to pore throat size ratio is large. The second mechanism called by-pass occurs when finite clusters of the non-wetting phase are formed by the advancing wetting phase due to multiple connections existing in the pore space that allow occupancy of the surrounding pores before displacing the cluster filling non-wetting fluid. Thus, trapping of the non-wetting phase occurs when a set of large pores is wholly surrounded by smaller pores. During imbibition, the wetting fluid will advance by occupying the pore-space from lower to higher pores sizes, but constrained by connectivity of both advancing and receding phases. When the occupancy for the advancing wetting phase allows a critical length scale l_(c), the non-wetting phase becomes disconnected, and no further removal is possible. In fact, the above-mentioned critical length scale l_(c) will represent the most resistant connection in the least resistant pathway, and is given by the smallest pore size within the set of largest pore sizes that form a sample spanning pathway as set forth in previously incorporated U.S. Pat. No. 7,221,158 to Ramakrishnan. This is discussed in more details below.

Transforming critical length scale l_(c) into critical relaxation time T_(2oc) via Eq. 1, an estimate is obtained for the lower bound of the residual non-wetting phase saturation S_(cr):

S _(cr)≈∫_(T) _(2oc) ^(T) ^(2b) g _(o)(T _(2o))dT _(2o).  (7)

This is a lower bound because it accounts only for the non-wetting phase that occupies pores with l>l_(c). In reality, the isolated clusters may contain pores smaller than l_(c). The above equation neglects some of the trapping that occurs before the wetting phase is allowed to occupy l_(c). Therefore, Eq. 7 is to be regarded as an approximation for S_(cr), and is satisfactory for lattices with large coordination numbers. See, Ramakrishnan, T. S, and Wasan, D. T., “Two-phase Distribution in Porous Media: An Application of Percolation Theory,” Int. J. Multiphase Flow, 12(3):357-388 (1986).

As will be now appreciated, percolation theory may be used to interpret NMR measurements. First, the relationship between the number probability density function n_(l)(l) and g_(l)(l) may be written as

$\begin{matrix} {{{n_{l}(l)} = \frac{{g_{l}(l)}/{V(l)}}{\int_{0}^{\infty}{{{g_{l}(l)}/{V(l)}}{l}}}},} & (8) \end{matrix}$

assuming that the volume of the pore V is determined by the single length scale l. A pore shape parameter ν can be introduced to quantify this relationship as

V(l)∝l ^(V).  (9)

This relation and values of ν for sinusoidal and rectangular pores are investigated in Ramakrishnan, T. S, and Wasan, D. T., “Two-phase Distribution in Porous Media: An Application of Percolation Theory,” Int. J. Multiphase Flow, 12(3):357-388 (1986). In addition, previously incorporated U.S. Pat. No. 7,221,158 states that the value of ν may vary from 0 to 3, depending on the pore shape. For example, for spherical pores, ν=3, neglecting the volume contained in the pore throats.

The proposed approximation of the in situ formation is based on the site percolation model, where the pore space is represented by the lattice of sites connected by bonds. See, Essam, J. M. “Percolation Theory,” Rep. Prog. Phys., 43:834-912 (1980) and Shante, V. K. S., and Kirkpatrick, S., “An Introduction to Percolation Theory,” Adv. Phys., 20:325-357 (1971). The sites are made accessible for the fluid with a fixed probability p, and the fluid cannot advance through a blocked site. The transport property of the percolation model is then characterized by the critical percolation probability p_(c), defining the maximum p, below which the probability of creating the connected path between the opposite sides of the lattice is zero, Frish, H. L., and Hammersley, J. M., “Percolation Processes and Related Topics,” J. Soc. Induct. Math., 11:894-918 (1963), i.e., the non-wetting phase is still immobile.

In terms of number probability function (see Eq. 8) and the critical length scale l_(c), the critical percolation probability p_(c) can be expressed as

p _(c)=∫_(l) _(c) ^(∞) n _(l)(l)dl.  (10) (10)

Substituting n_(l)(l) from Eq. 8 in Eq. 10 with the assumption from Eq. 9 and making variable transformation from l to T_(2o) using Eq. 1, an expression is obtained for the critical percolation probability via T_(2oc), spin-spin relaxation distribution g_(o)(T_(2o)), and pore geometry factor ν:

$\begin{matrix} {p_{c} = {\frac{\int_{T_{2{oc}}}^{T_{2b}}{{T_{2o}^{- v}\left( {T_{2b} - T_{2o}} \right)}^{v}{g_{o}\left( T_{2o} \right)}{T_{2o}}}}{\int_{T_{0}}^{T_{2b}}{{T_{2o}^{- v}\left( {T_{2b} - T_{2o}} \right)}^{v}{g_{o}\left( T_{2o} \right)}{T_{2o}}}}.}} & (11) \end{matrix}$

Similarly, if the percolation approach is applied to understand residual water saturation, it will be appreciated that the largest pore size l_(wc) from the set of smallest pores constituting the fraction equal to critical percolation probability p_(c) is of interest. Thus, Eq. 10 transforms to

p _(c)=∫₀ ^(l) ^(wc) n _(l)(l)dl  (12)

Thus, Eq. 11 for wetting phase percolation model will be

$\begin{matrix} {p_{c} = \frac{\int_{0}^{T_{2{wc}}}{{T_{2o}^{- v}\left( {T_{2b} - T_{2o}} \right)}^{v}{g_{o}\left( T_{2o} \right)}{T_{2o}}}}{\int_{0}^{T_{2b}}{{T_{2o}^{- v}\left( {T_{2b} - T_{2o}} \right)}^{v}{g_{o}\left( T_{2o} \right)}{T_{2o}}}}} & (13) \end{matrix}$

where T_(2wc) is a critical observed spin-spin relaxation time corresponding to the critical length scale l_(wc) (also see Eq. 6).

According to one embodiment of the invention, the spin-spin relaxation distribution g_(o)(T_(2o)) should account for the presence of clay. Clay tends to form a micro-porous coating at the surface of the pore or it may occur in clumps essentially not participating in the fluid flow. The common structure of this coating can be represented by a stack of platelets. Water is effectively trapped in the clay and cannot be displaced by another fluid under normal circumstances of displacement. Presence of clay in the pore structure affects the interpretation of NMR. In other words, given a volume density function, the number density function is highly skewed towards small pores. These small pores do not participate in percolation-like displacement and should therefore be discounted. One may perceive of them to add to the inventory of the wetting phase without affecting displacement processes.

Since pores classified as clays have a distinctly small size compared to the rest of the porous medium, a cut-off relaxation time T₂ below which only pores constituting clay are present is expected. This cut-off time labeled T_(2cb) corresponds to the largest pores within the clay and they do not see a phase replacement. It is therefore appropriate to compute “clay-bound” water from

S _(wb)=∫₀ ^(T) ^(2cb) g _(o)(T _(2o))dT _(2o)  (14)

where the commonly used value for T_(2cb) is 3 ms.

The clay contribution to the observed relaxation g_(o)(T_(2o)) is excluded from pore size distribution used to build the percolation model. Therefore, according to the embodiment, the clay-corrected relaxation distribution g_(c)(T_(2o)) accounting for the presence of clay should be used in Eq. 11 and Eq. 13 in lieu of g_(o)(T_(2o))

$\begin{matrix} {{g_{c}\left( T_{2o} \right)} = \left\{ \begin{matrix} {0,} & {T_{2o} \leq T_{2{cb}}} \\ {{{g_{o\;}\left( T_{2o} \right)}/{\int_{T_{2{cb}}}^{T_{2b}}{{g_{o}\left( T_{2o} \right)}{T_{2o}}}}},} & {T_{2o} \leq {T_{2{cb}}.}} \end{matrix} \right.} & (15) \end{matrix}$

Accounting for the presence of clay does not require revision of Eq. 7 for residual CO₂ saturation since CO₂ will be trapped in the pores corresponding to relaxation times T_(2o)>T_(2cb), and S_(cr) is calculated as a ratio of the pore-space within which trapping has occurred to the total porosity of the formation. The estimate for residual water saturation S_(wr) (Eq. 6) also holds, since both clay-bound water and residually trapped water contribute to S_(wr). Thus, it is only for the calculation of T_(2oc) and T_(2wc) that g_(c)(T_(2o)) from Eq. 15 is used in the embodiment which accounts for the presence of clay.

Using the theoretical analysis set forth above, the formation may be characterized utilizing NMR data obtained from a borehole tool such as the Combinable Magnetic Resonance tool (CMR—a trademark of Schlumberger), and the MR Scanner tool—a trademark of Schlumberger. The specific method of finding the residual carbon dioxide saturation depends upon the available information which can be used to calibrate the percolation model as described hereinafter. In addition, as will be set forth hereinafter, even in the absence of other petrophysical data, a realistic range for the S_(cr) values may be obtained, and this range can be reduced to a single best estimate if more data to calibrate the underlying percolation model become available. Furthermore, the methods described herein allow consistent propagation of uncertainty in the parameters of the model to characterize uncertainty in the resultant estimation for S_(cr).

Turning to FIG. 1 a first embodiment of a method for estimating residual carbon dioxide values is provided. The first embodiment relates to a situation where only magnetization relaxation data is available. Using an NMR tool (as described in more detail hereinafter with reference to FIG. 4) such as the Schlumberger CMR or CMR-plus tool, for every depth of interest, the transverse magnetization relaxation over time M(t) is measured at 110. Then, utilizing computational procedures such as disclosed in U.S. Pat. No. 5,363,041, the relaxation distribution g_(o)(T_(2o)) is calculated at 120. At 130, the relaxation distribution is optionally corrected for clay content according to Eq. 15. In order to calculate T_(2oc) from the (clay corrected) relaxation distribution, values for critical percolation probability p_(c) and the pore geometric factor ν are required. In the absence of other relevant data, a range for the values of critical percolation probability p_(c) is assumed to be between 0.2 and 0.3, while that for the pore geometric factor ν is assumed to be between 0 and 3. However, if additional data on pore shapes can be inferred from other sources (e.g., core data using CT scans), the default range of values for ν may be narrowed. Regardless, the uncertainty in available knowledge about p_(c) and ν may be represented via a probability distribution function (e.g., a normal, lognormal, triangular, uniform or other such probability density function) at 140, and random sampling can be used to generate an i^(th) pair of values taken at 150. It will be appreciated that Latin Hypercube Sampling (Helton, J. C. and Davis, F. J., “Latin Hypercube Sampling and the Propagation of Uncertainty in Analyses of Complex Systems,” Reliab Eng Syst Saf, 81:23-69 (2003)) may provide an efficient manner if generating equiprobable samples from a set of variables defined by their probability distributions. The pair of values is then used in conjunction with the calculated (clay corrected) relaxation distribution to calculate T_(2oc) according to Eq. 11 (optionally in light of Eq. 15). From the calculated T_(2oc), the residual carbon dioxide saturation S_(cr) can be estimated at 170 utilizing Eq. 7.

As indicated in FIG. 1, because a single pair of values p_(c) and ν were chosen at 150 to be used in calculating T_(2oc) at 160, it is desirable to repeat steps 150-170 a plurality (e.g. N>100) times. Each time, a new pair of values for p_(c) and ν are randomly generated, and used to calculate T₂ and S_(cr). As a result, a probability density function (pdf) of the residual carbon dioxide saturation can be generated for each depth at which NMR data was collected. Alternatively, or in addition, a single best estimate can be calculated as the mean or mode value of the obtained S_(cr) distribution. The results can be generated as a simple log of most likely values as a function of depth (as seen in FIG. 5 a described below), or as a log of pdf values as a function of depth using intensity or color (as seen in FIG. 5 b described below).

It is noted that if a range of values is specified for p_(c) and ν, only two iterations are needed to calculate a range of possible values for S, since S_(cr) monotonously increases with increasing ν and increasing p_(c). Therefore, steps 160 and 170 can be performed only for two pairs of percolation parameters: (min (p_(c)); min (ν)) and (max (p_(C)); max (ν)), thus generating the bounds for S_(cr) for any given depth.

Turning to FIG. 2, another embodiment of a method for estimating residual carbon dioxide values is provided. This embodiment relates to a situation where residual water saturation (S_(wr)) information is obtained at 205 in addition to magnetization relaxation data obtained at 210. It will be appreciated that the residual water saturation may be obtained at 205 from any of several mechanisms including but not limited to core analysis, or scanning using a nuclear logging tool during injection of carbon dioxide into the formation. Regardless, steps 210-230 are identical to steps 110-130 of FIG. 1 where the magnetization relaxation data M(t) obtained from an NMR measurement are transformed into a relaxation distribution (or normalized into a probability density function) g_(o)(T_(2o)) and corrected for the presence of clay. Given the estimate for S_(wr) at 235, Eq. 6 is solved for T_(2wc).

If an estimate for S_(wr) is available at 205, only one of the two parameters of the percolation model has to be “guessed” as the other one can be calculated. Since p_(c) has a narrower range of possible values compared to that of ν, and dependence of S_(cr) on p_(c) in this range is almost linear, in one embodiment p_(c) is selected as the parameter to be guessed or assigned a range of possible values (although in another embodiment it will be appreciated that ν may be selected as the parameter to be guessed or assigned a range of possible values). At 240, the uncertainty in the estimate of p_(c) can be represented by probability distribution. N equiprobable values of p_(c) can then be randomly generated at step 250 using Latin Hypercube Sampling or any other method of random sampling honoring the probability distribution. Then, utilizing values for T₂, and p_(c), the pore geometry factor ν can be calculated at step 260 utilizing Eq. 13, and from the pore geometry factor and the previously calculated (clay-corrected) relaxation distribution (and the same realization for p_(c) generated at step 250), T₂ is calculated at step 270 according to Eq. 11 (optionally in light of Eq. 15). From the calculated T₂ and the spin-spin relaxation time distribution g_(o)(T_(2o)), the residual carbon dioxide saturation S_(cr) can be estimated at 280 utilizing Eq. 7.

As in the method of FIG. 1 where steps 150-170 were repeated a plurality of times, in the method of FIG. 2, steps 250-290 are performed a plurality (N) times, and at the end of the N-th iteration, N values of S_(cr) are generated. The probability distribution for values of S_(cr) can be calculated at step 290 and presented as a log. In other words, a probability density function (pdf) of the residual carbon dioxide saturation can be generated for each depth at which NMR data was collected. Alternatively, or in addition, a single best estimate can be calculated as the mean or mode value of the obtained S_(cr) distribution. The results can be generated as a simple log of most likely values as a function of depth (as seen in FIG. 5 a described below), or as a log of pdf values as a function of depth using intensity or color (as seen in FIG. 5 b described below).

According to another embodiment of the invention, if a single value for S_(wr) is not available, but it is possible to obtain a range of from the available petrophysical data or estimate a probability distribution on this range, in lieu of step 205, step 250 can be modified to generate equiprobable realizations of pairs (p_(c); S_(wr))_(i). Then, step 235 will be performed at every iteration and precede step 260.

If only maximum and minimum values are specified for p_(c) at step 240, only two iterations are needed to calculate a range of possible values for S_(cr), since S_(cr) monotonously increases with increasing p_(c). If only minimum and maximum estimates for S_(wr) are available, the range of possible values of S could be estimated based on two iterations for pairs (min (p_(c)), max (S_(wr))) and (max (p_(c)), min (S_(wr))).

Turning to FIG. 3, yet another embodiment of a method for estimating residual carbon dioxide values is provided. This embodiment relates to a situation where a permeability estimate k is obtained at 305 in addition to magnetization relaxation data obtained at 310. The permeability estimate may be obtained utilizing injection or drawdown measurements from a formation tester such as the MDT (a trademark of Schlumberger) or in any other desired manner and is used to calculate T_(2oc) as discussed below. In FIG. 3, steps 310, 320, and 330 are the same as steps 110-130 and 210-230 discussed above with reference to FIGS. 1 and 2 and translate M(t) to obtain g_(o)(T_(2o)) and g_(o)(T_(2o)) corrected for the presence of clay.

It will be appreciated that based on percolation theory, and as set forth in previously incorporated U.S. Pat. No. 7,221,158 to Ramakrishnan, in situ formation permeability is related to the critical relaxation time T_(2oc) according to:

$\begin{matrix} {k = {\frac{\varphi^{m}}{8}\rho^{2}\frac{T_{2{oc}}^{2}T_{2b}^{2}}{\left( {T_{2b} - T_{2{oc}}} \right)^{2}}}} & (16) \end{matrix}$

where φ is the porosity (preferably obtained from nuclear or density measurements corrected by an elemental spectroscopy tool as known in the art), m is a constant, and ρ is the surface relaxivity. In order to solve Eq. 16, the value of surface relaxivity ρ is required. It is known that possible values of ρ for sandstone formations can be as low as 4-8 μm/s in Fontainebleau sandstones and as high as 20-26 μm/s in Berea sandstones. In the absence of information about the type of the sandstone formation in hand, one recommended value for ρ is 16 μm/s. Regardless, a probability density function for ρ is generated at 340 from which samples can be generated at 350. Given the values for permeability k, the surface relaxivity ρ, and the bulk fluid relaxation time T_(2b), Eq. 16 can be solved at step 360 for T_(2oc) (with m=2) according to:

$\begin{matrix} {T_{2{oc}} = {\frac{2\sqrt{2k}}{{\varphi \; \rho \; T_{2b}} + {2\sqrt{2k}}}T_{2b}}} & (17) \end{matrix}$

From the calculated T_(2oc), the residual carbon dioxide saturation S_(cr) can be estimated at 370 utilizing Eq. 7.

As in the method of FIG. 1 where steps 150-170 were repeated a plurality of times, in the method of FIG. 3, steps 350-380 are performed a plurality (N) times, and at the end of the N^(th) iteration, N values of S_(cr) are generated. The probability distribution for values of S_(cr) can be calculated at step 380 and presented as a log. In other words, a probability density function (pdf) of the residual carbon dioxide saturation can be generated for each depth at which NMR data was collected. Alternatively, or in addition, a single best estimate can be calculated as the mean or mode value of the obtained S_(cr) distribution. The results can be generated as a simple log of most likely values as a function of depth (as seen in FIG. 5 a described below), or as a log of pdf values as a function of depth using intensity or color (as seen in FIG. 5 b described below).

It should be appreciated that for values of T_(2b)=3 s, φ=0.2, and k=100 mD, Eq. 17 results in a value of T_(2oc)=0.256 s for ρ=16 μm/s, while the range of ρ from 8 μm/s to 24 μm/s, would produce values for T_(2oc) from 0.471 s to 0.175 s respectively. With increasing T_(2oc) the estimate for S_(cr) decreases (see Eq. 7). The actual amount of decrease will depend on g_(o)(T_(2o)).

The estimated value for T_(2oc) obtained at step 360 can be also used to infer a value for the pore geometry factor ν. Since the value of ν is not expected to change significantly within the same stratigraphic layer, a single measurement of the permeability k in this layer can provide enough information to calibrate the percolation model for an estimation of S_(cr) based on NMR measurements throughout the layer. More particularly, certain steps of FIG. 3 should be expanded to allow for an estimation of ν. Specifically, step 340 could be expanded to include step 345 that provides a probability distribution for the critical percolation probability p_(c) value. Thus, step 350 can now be replaced by step 355, where for every iteration, the equiprobable realization of the pair (p_(c); ρ)_(i) is generated. Once the value for T_(2oc) is obtained at step 360, an estimate for the pore geometric factor ν can be calculated according to Eq. 11 at step 375.

After N iterations, the conditional probability density function for ν can be generated at step 385. Hence, a station measurement of permeability may be used to infer a conditional ν distribution, thus allowing the generation of the S_(cr) distribution throughout a stratigraphic layer. Then a continuous log of S_(cr) may be generated. It will be appreciated that a bivariate probability density function for (p_(c),ν) can also be generated from this embodiment and used in the embodiment of FIG. 1 in lieu of the (p_(c),ν) samples generated at step 150.

A logging apparatus for carrying out the previously-described methods is shown schematically in FIG. 4. More particularly, a tool 410 is shown suspended in a borehole 412 traversing a formation 414 from the lower end of a typical multiconductor cable 415 that is spooled in a usual fashion on a suitable winch (not shown) on the formation surface. On the surface, the cable 415 is preferably electrically coupled to an electrical control/processor system 418. The tool includes an elongated body 419 which encloses the downhole portion of the tool control system 416. The elongated body 419 includes an NMR assembly or module 420 (such as a CMR or a CMR-Plus type tool or MR-Scanner) typically including magnets (not shown), an antenna (not shown) and electronics (not shown). The tool also optionally carries an MDT-type module 423 including a selectively extendable fluid admitting assembly 421 (where injection or drawdown is desired), a selectively extendable tool anchoring member 422 which may be arranged on an opposite side of the body from the fluid admitting assembly, and a fluid analysis module (not shown) through which the obtained fluid flows. With the provided tool, the electronics 418 on the surface (e.g., properly programmed computer/processor apparatus) may be used to control the tool 410 such that NMR measurements are obtained and processed or pre-processed downhole and forwarded to the surface for further processing based on the previously described methods. The results of the processing by the uphole processing system 418 are preferably one or more logs of residual carbon dioxide saturation as a function of formation depth.

One example of an output of processing system 418 is seen in a fabricated FIG. 5 a where the residual carbon dioxide saturation determination (estimation) is shown as a function of formation depth, i.e., as a log. The log may be provided on paper or may be displayed on a monitor or screen. The residual carbon dioxide saturation log may be used as an input into the reservoir model to predict performance of the storage site including predictions for CO₂ plume migration during and after injection, and uncertainty analysis in the performance of the storage site, as disclosed in U.S. Ser. No. 12/768,021 to Chugunov et al., entitled “Method for Uncertainty Quantification in the Performance and Risk Assessment of a Carbon Dioxide Storage Site,” which is hereby incorporated by reference herein in its entirety.

Another example of an output of processing system 418 is seen in a fabricated FIG. 5 b where the residual carbon dioxide saturation determination (estimation) is shown as a probability density function (pdf) as a function of formation depth. The pdf may be shown in different manners, e.g., by color or intensity. Thus, as seen in FIG. 5 b, a gray scale is used to show the likelihoods of the S value at different locations in the formation. While the log of FIG. 5 b is not shown to be smooth, it will be appreciated by those skilled in the art that depending upon the granularity of the data and depending upon the use of smoothing filters, estimations, etc., FIG. 5 b could be modified so that the percentile probabilities would be smooth from one depth to another along the formation. It will be appreciated that the log of FIG. 5 b could be used for the same purposes as the log of FIG. 5 a.

There have been described and illustrated methods and apparatus for determining residual carbon dioxide saturation in a formation utilizing NMR measurements. While particular embodiments of the invention have been described, it is not intended that the invention be limited thereto, as it is intended that the invention be as broad in scope as the art will allow and that the specification be read likewise. Thus, while particular tools have been described for obtaining nuclear magnetic resonance measurements as well as indications of formation characteristics such as permeability, it will be appreciated that other tools may be utilized provided they function to obtain equivalent measurements. In addition, while particular equations have been described as establishing relationships between certain measurements and formation properties, it will be understood that other similar equations could be utilized. Also, while certain values and/or ranges were described for certain variables of the equations, and certain techniques for selecting samples of those values were described it will be recognized that other suitable values and ranges can be used and other techniques for generating sample values utilized. It will therefore be appreciated by those skilled in the art that yet other modifications could be made to the provided invention without deviating from its spirit and scope as claimed. 

What is claimed is:
 1. A method of calculating critical cut-off transverse relaxation time T_(2oc) for a formation, comprising: obtaining an indication of permeability k for the formation at a depth; and calculating a critical relaxation time T_(2oc) according to ${T_{2{oc}} = {\frac{2\sqrt{2k}}{{\varphi \; \rho \; T_{2b}} + {2\sqrt{2k}}}T_{2b}}},$ wherein ρ is the surface relaxivity at the depth, φ is porosity at the depth, and T_(2b) is bulk transverse relaxation time of the fluid.
 2. The method of claim 1, further comprising making NMR measurements at the depth in the formation to obtain a determination of transverse magnetization decay M(t) at the depth, and determining a relaxation distribution go(T_(2o)) from M(t) using a processor.
 3. The method of claim 2, further comprising calculating an estimate of residual CO₂ saturation S_(cr) according to S_(cr)≈∫_(T) _(2oc) ^(T) ^(2b) g_(o)(T_(2o))dT_(2o).
 4. The method of claim 2, wherein said relaxation distribution is a clay-corrected relaxation distribution.
 5. The method of claim 3, further comprising displaying an indication of the estimate of the residual carbon dioxide saturation.
 6. The method of claim 2, wherein the relaxation distribution is determined according to M(t)=φ∫₀ ^(T) ^(2b) g_(o)(T_(2o))exp(−t/T_(2o))dT_(2o), where g_(o)(T_(2o)) is said relaxation distribution, T_(2b) is the bulk transverse relaxation time of fluid at the depth, and φ is porosity at the depth.
 7. The method of claim 3, wherein the plurality of residual CO₂ saturation values are determined by utilizing a plurality of different pairs of values for k and ρ.
 8. The method of claim 2, further comprising estimating a pore geometry factor ν for a geological layer using estimated value of T_(2oc) by solving the equation ${p_{c} = \frac{\int_{T_{2{oc}}}^{T_{2b}}{{T_{2o}^{- v}\left( {T_{2b} - T_{2o}} \right)}^{v}{g_{o}\left( T_{2o} \right)}{T_{2o}}}}{\int_{T_{0}}^{T_{2b}}{{T_{2o}^{- v}\left( {T_{2b} - T_{2o}} \right)}^{v}{g_{o}\left( T_{2o} \right)}{T_{2o}}}}},$ wherein p_(c) is the critical percolation probability.
 9. The method of claim 8, wherein the plurality of pore geometry factor values ν is determined by utilizing a plurality of different pairs of values for p_(c) and ρ.
 10. The method of claim 9, wherein the utilizing a plurality of different pairs of values for p_(c) and ρ comprises assuming probability density functions for p_(c) and ρ, and selecting pairs randomly from the probability density functions.
 11. The method of claim 8, further comprising using the estimated value of pore geometry factor ν for a geological layer to calculate the value of T_(2oc) at a second depth within the layer by solving the equation ${p_{c} = \frac{\int_{T_{2{oc}}}^{T_{2b}}{{T_{2o}^{- v}\left( {T_{2b} - T_{2o}} \right)}^{v}{g_{o}\left( T_{2o} \right)}{T_{2o}}}}{\int_{T_{0}}^{T_{2b}}{{T_{2o}^{- v}\left( {T_{2b} - T_{2o}} \right)}^{v}{g_{o}\left( T_{2o} \right)}{T_{2o}}}}},$ and using estimated value of p_(c) and calculated relaxation distribution function g_(o)(T_(2o)) at the second depth within the geological layer.
 12. The method of claim 11, further comprising calculating S_(cr) according to S_(cr)≈∫_(T) _(2oc) ^(T) ^(2b) g_(o)(T_(2o))dT_(2o).
 13. The method of claim 12, wherein the plurality of residual CO₂ saturation values S_(cr) is determined by utilizing a plurality of different pairs of values for p_(c) and ρ.
 14. A system for making residual carbon dioxide saturation estimates for a formation traversed by a borehole, comprising: a) a nuclear magnetic resonance tool capable of traversing the borehole and making nuclear magnetic resonance measurements at a depth in the formation to obtain a determination of transverse magnetization decay M(t) at that depth; b) a processor coupled to the nuclear magnetic resonance tool, wherein the processor, 1) determines a relaxation distribution from the M(t), 2) estimates a critical relaxation time T_(2oc), wherein the critical relaxation time T_(2oc) is calculated according to $T_{2{oc}} = {\frac{2\sqrt{2k}}{{\varphi \; \rho \; T_{2b}} + {2\sqrt{2k}}}T_{2b}}$ where ρ is the surface relaxivity at the depth, k is permeability, and φ is porosity at the depth, and 3) utilizes T_(2oc) to determine an estimate of the residual carbon dioxide saturation S_(cr) at the depth in the formation; and c) a display that displays an indication of the estimate of the calculated residual carbon dioxide saturation.
 15. The system of claim 14, wherein the relaxation distribution is a clay-corrected relaxation distribution, and the processor further determines a plurality of critical relaxation times for the depth and determines a plurality of estimates of the residual carbon dioxide saturation.
 16. The system of claim 15, wherein the display shows a probability density function, a mode value, a mean value, a value range for the residual carbon dioxide saturation from the plurality of estimates, or a combination thereof.
 17. The system of claim 16, wherein the display is a function of depth in the formation.
 18. The system of claim 17, wherein the processor determines the relaxation distribution according to M(t)=φ∫₀ ^(T) ^(2b) g_(o)(T_(2o))exp(−t/T_(2o))dT_(2o), where g_(o)(T_(2o)) is the relaxation distribution, T_(2b) is the bulk transverse relaxation time of fluid at the depth, and φ is porosity at the depth, and the estimate of S_(cr) according to one of S_(cr)≈∫_(T) _(2oc) ^(T) ^(2b) g_(o)(T_(2o))dT_(2o) and S_(cr)≈∫_(T) _(2oc) ^(T) ^(2b) g_(c)(T_(2o))dT_(2o), where g_(c)(T_(2o)) is the clay-corrected relaxation distribution. 